Prove that from lim xn = a it follows that lim |xn| = lal. Is the converse true?

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Okay, so firstly we want to show that if the limit of X N equals A then the limit of the absolute values of X. N equals the absolute value of A. So we're going to let Episode one Be greater than zero and choose. And in N. Such that for all and greater than or equal to n X n minus A is less than epsilon. Which we can do. Since we know that this limit exists, then we know that for all N greater than or equal to N. So what do we want to show? We want to show that? So let me delete this. We want to show that uh for all and greater than or equal to N, the difference between the absolute value of X. N and the absolute value of A is less than epsilon. So if we can show this then we have this converges to this. But we know that xn minus a. An absolute value like this is less than or equal to x N minus A. And this is by the reverse triangle inequality. The reverse triangle inequality we have that the absolute value of xn minus the absolute value of a. All in absolute value is less than or equal to the absolute value of xn minus A. And we already know that this thing here is less than epsilon. Since that's how we chose our end above. So we have that for all epsilon greater than zero, there exist N N. N. Such that for all N greater than or equal to n X N absolute value minus the absolute value of a is less than epsilon. So we've shown that the limit as X. And the limit as N tends to infinity of the absolute value of X n is equal to the absolute value of a. Now we want to look at the converse. So let's suppose we have a sequence. So that's that part done. Now, I suppose we have a sequence such that the limit as N tends to infinity of the absolute value of X N is equal to the absolute value of A. And we want to ask does the limit as N tends to infinity of xN equal A. And the answer is no. So let's provide a counter example. So let's look at the sequence XN equals -1 to the end. Okay, so let's firstly do this part show that the limit of the absolute value of accent exists. So the absolute value of X n is just one, right? Because this is either one or minus one depending on if N is odd or even. So the absolute value of X n is one for all N. And this converges obviously to one as N tends to infinity. So we have the absolute value of XN converges to one. But the sequence xn doesn't actually converge. So X n is minus one to the N. Which is the sequence minus 11 minus 11 minus 11, et cetera. So -1N doesn't converge. So we find a sequence which converges if you just take absolute values, but the sequence itself doesn't converge. So this is not true this implication.

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